Muckenhoupt type weights and Berezin formulas for Bergman spaces

Abstract

By means of Muckenhoupt type conditions, we characterize the weights ω on $\mathbb{C}$ such that the Bergman projection of $F_{\alpha }^{2,\ell }=H(\mathbb{C})\cap L^2(\mathbb{C},e^{-\frac{\alpha }{2}|z{|}^{2\ell }})$, $\alpha >0$, $\ell >1$, is bounded on $L^p(\mathbb{C},e^{-\frac{\alpha p}{2}|z{|}^{2\ell }}\omega (z))$, for $1<p<\infty$. We also obtain explicit representation integral formulas for functions in the weighted Bergman spaces $A^p(\omega )=H(\mathbb{C})\cap L^p(\omega )$. Finally, we check the validity of the so called Sarason conjecture about the boundedness of products of certain Toeplitz operators on the spaces $F_{\alpha }^{p,\ell }=H(\mathbb{C})\cap L^p(\mathbb{C},e^{-\frac{\alpha p}{2}|z{|}^{2\ell }})$.